Structural behaviour of fixed ended stocky Lean Duplex Stainless Steel (LDSS) flat oval hollow column under axial compression

Thin-Walled Structures 113 (2017) 47–60

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Structural behaviour of fixed ended stocky Lean Duplex Stainless Steel (LDSS) flat oval hollow column under axial compression

MARK

⁎

Khwairakpam Sachidananda, Konjengbam Darunkumar Singh Department of Civil Engineering, Indian Institute of Technology Guwahati, India

A R T I C L E I N F O

A B S T R A C T

Keywords: Lean duplex Abaqus Flat-oval Slender AS/NZS 4673 EN 1993-1-4 Direct strength method

This paper presents finite element analyses of Lean Duplex Stainless Steel (LDSS) flat oval hollow slender columns of stocky cross sections, by varying geometrical parameters like flat element length (lf), radius of the curve element (r), distance between the flat elements (w), length/height of the columns (l), and thickness (t), using Abaqus. Based on the FE analyses, structural performance of the columns under axial compression were assessed, with attention towards, load capacity and deformation modes. Based on the parametric study, failure or deformation mechanisms have been identified depending upon ultimate and post ultimate structural behaviour viz., 1) Yield to yield with local bucking (λ < 0.32, 2) Yield to yield with flexural buckling (0.32≤λ ≤1.0), and 3) Flexural to flexural buckling (λ > 1.0). It is seen that semi-circular curve section (r/w=0.5) showed the highest load capacity, with Pu remaining almost flat for r/w≥1.0. Load capacity, in general increases with increase in lf, w, and t. Pu decreases with l, although, the rate of decrease is relatively less sensitive to variations in lf and r; however, it is quite sensitive to variation in w. All the compared design codes, method and their modified versions are reliable and applicable, with βo value of 3.28, 3.52, 3.90 for EN-1993-1-4 (2015), AS/NZS 4673 (2001) and DSM (NAS [38]); and 3.38, 3.50 for modified AS/NZS and modified DSM (Huang and Young [9]) respectively.

1. Introduction As early as the late 19th century, 'carbon' steel became a popular construction material choice and began to influence the construction industry (TataSteelConstruction.com) [1]. Since then, a prevalent use of carbon steel especially as structural members can be seen till date. Amongst the various reasons for its popularity with the engineers and architects in the recent decades, mention can be made of e.g. low cost, easy availability, long experience, different strength grades, well developed design rules, etc. However, its inherent low corrosion resistance became an obvious major drawback, which can significantly subside its durability, notably for corrosive exposures. But with the invention of stainless steel, a significant appealing improvement over carbon steel, could be seen, e.g., higher strength to weight ratio, good corrosion resistance, low maintenance cost, high ductility, impact resistance, fire resistance, durability, recyclability (100% recyclable with no degradation) etc. Further stainless steel can provide smooth, uniform and glossy finish, resulting in excellent aesthetic appearance. This led a boost in the utilization of stainless steel as structural members, particularly in exposed architectural designs. From an economy perspective, it has been estimated that

⁎

stainless steel can offer a cost-effective option as compared to carbon steel, if a whole-life costing method is adopted. In spite of its attractive advantages, the use of stainless steel in the construction industry remains relatively limited as compared to that of carbon steel, primarily owing to the need of specialist expertise of its structural behaviour (e.g. Mann [2], Gardner [3], Ashraf et al. [4], Theofanous and Gardner [5]). Predominantly, in the construction industry, stainless steel of the austenitic variety are commonly used. However, due to its high nickel content (~8–10% by mass), the cost of austentic variety remains high (~3–5 times that of carbon steel). However, a new breed of stainless steel named Lean Duplex Stainless Steel (LDSS) has been introduced offering striking benefits such as cheaper cost (nickel content=~1.5% i.e. much lower as compared to austenitic stainless steel), higher strength, acceptable weldability and fracture toughness properties, improved high temperature properties, etc. (e.g. Theofanous and Gardner [5], Saliba and Gardner [6]). As a result, in the recent times, several researchers (e.g. Gardner and Young and their collaborators viz., Theofanous and Gardner [5], Gardner and Nethercot [7], Saliba and Gardner [6], Huang and Young [8,9] etc). In the literature, several studies have been reported related to effect of both open and closed cross sections on the structural behaviour of

Corresponding author. E-mail address: [email protected] (K.D. Singh).

http://dx.doi.org/10.1016/j.tws.2017.01.012 Received 23 September 2016; Received in revised form 25 December 2016; Accepted 9 January 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.

Thin-Walled Structures 113 (2017) 47–60

K. Sachidananda, K.D. Singh

r (curvature radius), lf (flat length), t (thickness), w (width between flat plates maintaining semi-circle curve portion at the ends) and l (column height). The FE analyses have been performed considering both the material non-linearity and geometric (local and global) imperfections. The first part of the study, involves the validation of the FE modeling approach by comparing with experimental results of LDSS square hollow section columns under axial compression, of different lengths [13]. After the validation of the current FE modeling approach, parametric study of the fixed ended LDSS flat oval hollow columns are presented. The basic FE modeling approaches adopted in the present study are summarized in the following sub-sections. It may be mentioned that the present FE modeling approach considered here is in line with those published widely in the literature for similar type of studies i.e. closed or opened steel members under compression (e.g., Zhu and Young [21] for cold-formed flat-oval sections, Chan and Gardner [39] for hot-rolled elliptical sections, Gardner and Ministro [18] for oval sections, Patton and Singh [40] for LDSS L, T, and + sections, Sachidananda and Singh [19] for LDSS flat-oval sections, Saliba and Gardner [11] for LDSS I-sections etc).

tubular steel (both carbon and stainless steel) columns e.g. I-section (Ashraf et al. [10]; Saliba and Gardner [11]), Circular Hollow section (CHS) (Ashraf et al. [12]), Square Hollow Section (SHS) (Theofanous and Gardner [13]; Gardner [14], Rectangular Hollow Section (RHS) (Theofanous et al. [15]), L, T, + (Patton and Singh [16]) etc. Lately, aesthetically pleasing cross sections such as Elliptical Hollow Section (EHS) (Silvestre [17]) and Oval Hollow Section (OHS) (Gardner and Ministro [18];, Sachidananda and Singh, [19]) have gained research focus. In addition to its unique aesthetically pleasing shapes, these cross section furnish variant cross sectional properties with regard to both minor and major axes (Zhu and Young [20,21]). Studies on elliptical/oval hollow section loaded with axial compression have been published as 1950s, primarily due to the requirements (seeking improved strength to weight ratios) of both aerospace and automobile industries e.g. Hutchinson [22], Kemper and Chen [23], Feinstein et al. [24] reported on the elastic buckling of elliptical/oval hollow through analytical studies; Parks and Yu [25] presented on both analytical and experimental studies on the interaction effect of curved and flat elements of stiffened flat oval (having both flat and curved elements connected through bolting) steel stub columns. For civil engineering applications (as structural elements), few researches have been reported on cold formed oval hollow steel sections e.g. Gardner and Ministro [18], Theofanous et al. [26], Zhu and Young [20,21], Silvestre and Gardner [27]. In this context, it is worthwhile to mention that, probably, Zhu and Young [20,21] were the first to initiate work on cold formed flat oval hollow steel section (a single section made of two flat web and two rounded semi-circular flange faces) columns under compressive load using both experimental and non-linear finite element buckling analyses, for potential application towards civil engineering structures. An extension of such studies has also been reported by Zhao and Packer [28] and Yang et al. [29] on concrete-filled oval steel columns. Structural behaviour of concrete filled flat oval (with rounded ends) steel tubular stub columns under axial compression has been presented by Wang et al. [30]. In the construction industry, use of such oval/elliptical sections can be seen in places such as at Legends center, Canada, Electronics arts stairwell, Canada [31], Barajas airport, Spain; Heathrow airport, UK; Cork airport, Ireland; Zeeman building, University of Warwick, UK, Society bridge, Scotland [32] indicating confidence amongst engineers and architects for such new sections. Examples of flat oval steel sections that are available commercially are Form 220 and 370 flat oval sections from Ruukki [33]. However, it must be noted that the current steel design codes do not have exclusive provisions for flat oval hollow sections, possibly due to the lack of sufficient information on its structural performances. Thus, the aim of the current study is to investigate systematically the effects of cross-sectional parameters viz., element radius, flat element length, flat elements separation distance, thickness, and column lengths, on the structural behaviour (e.g. load and deformation capacities) of axially loaded LDSS flat oval hollow columns of 'stocky' sections, through finite element (FE) analyses using Abaqus [34]. This study serves as an extension of an earlier study on LDSS hollow 'stub' columns under pure axial load, reported by the authors (Sachidananda and Singh [19]). Based on the FE results presented herein, the applicability of EN 1993-1-4 [35], AS/NZS 4673 [36] and DSM [37,38] are assessed. Further, comparison of modified AS/NZS 4673 [9] and DSM [9] for LDSS by Huang and Young [9] with those of present FE results are also presented.

2.2. Geometry and boundary conditions Figs. 1 and 2 show the schematic representation of a typical crosssection and geometry, meshing, boundary conditions employed in the FE models. In order to observe the effect of several cross-sectional components viz., curvature radius (r), flat length (lf), and thickness (t), broad ranges of their values (25–100 mm, 50–150 mm, 10–20 mm respectively) have been considered. It may be noted that based on the flat element cross-section (which is a weaker section compared to the curved section), these thickness values corresponds to the Class 1 (lf/ ⎛ 235 E ⎞0.5 tε≤33, where ε = ⎜ f 210000 ⎟ ; E and fy are Young's modulus and yield ⎠ ⎝ y stress respectively) and Class 2 sections (lf/tε≤35) i.e. stockier/thicker sections (lf/tε≤35) as per EN 1993-1-4 [35]. For covering the entire spectrum of length conditions (stub as well as slender lengths), 1000– 3500 mm lengths of columns have been examined. In the FE models, the column is being idealized as being fixed at the bottom and loaded at the top end. The loaded top end part is allowed to translate freely along the axial direction of the column. The boundary conditions have been achieved through the adoption of Reference Points (RP-1 and RP-2), which constraints the column ends via kinematic coupling available in Abaqus [34], as depicted in Fig. 2. All the degrees of freedom of the nodes at the loaded end are restrained (through RP2), except for the translation along the column axis direction. At the reference point, RP2, normal (along the column axis) displacement was applied using displacement control regime. 2.3. Finite element mesh The column models were discretised with S4R elements (four-noded doubly curved shell element having six degrees of freedom per node with reduced integration scheme) available in Abaqus [34]. The S4R element is reported widely in the literature to provide reasonably

2. Finite element modeling 2.1. General In order to assess the structural behaviour of fixed ended LDSS flat oval hollow column, finite element analyses have been conducted considering variation of the geometric parameters of the column viz.,

Fig. 1. Flat oval hollow section.

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by Gardner and Ashraf [43] for LDSS material consists of two distinct parts viz., i) Ramberg-Osgood model up to 0.2% proof stress (σ0.2), and ii) Gardner and Ashraf model from σ0.2 to σ1.0 ( see Eq. (1)) n 0.2,1.0 ⎛ σ − σ0.2 ⎞ ⎛ σ − σ0.2 ⎞ ⎛ σ − σ0.2 ⎞ ′ + εt 0.2 ε=⎜ ⎟ ⎟x ⎜ ⎟ + ⎜εt1.0 − εt 0.2 − 1.0 E0.2 ⎠ ⎝ σ1.0 − σ0.2 ⎠ ⎝ E0.2 ⎠ ⎝

(1) where єt1.0 and єt0.2 are total strains at σ1.0 and σ0.2, respectively; and n'0.2,1.0 is the strain hardening exponent. The material model proposed by Gardner and Ashraf [43] has been reported to provide reasonably good agreement with both under compression and tension loading conditions [16]. In the present material modeling, both the flat and corner portions of the hollow sections have the same material properties i.e. strength enhancement at the corner region (due to work hardening resulting from cold forming) [26,47] has been ignored. Thus the current FE approximation is likely to provide conservative values when compared to the actual experimental performance. Further, the effect of residual stresses has not been considered in the FE models, in line with similar studies in the literature [26,48]. It was observed by Huang and Young [9] that the effect of FE results between models with and without incorporation of measured membrane residual stresses (for cold formed LDSS columns) to be very small (~1%). Ellobody and Young [49] also reported similar findings for their work on cold formed high strength stainless steel columns.

Fig. 2. Typical FE (a) geometry, (b) FE mesh and (c) boundary conditions of LDSS flat oval hollow column.

accurate results for the analyses of both thick and thin hollow steel columns of various cross-sections such as square [13,16], oval [18,26], elliptical [17,27], etc. Based on linear elastic eigen value buckling analysis, an optimum FE element size of ~10 mm×10 mm has been arrived at through mesh convergence study. In the FE models, the total number of S4R elements ranges from ~2600 to 12600, maintaining element aspect ratio close to 1. Fig. 2(b) presents a typical S4R FE mesh for the LDSS oval hollow column.

2.6. Finite element model validation In order to validate the FE modeling approach adopted herein in this work, the experimental results reported by Theofanous and Gardner [13], on slender SHC (Square Hollow Column-80×80×4 and 60×60×3) of lengths, l=800, 1200, 1600 and 2000 mm, have been compared. The dimensions of the columns considered for the comparison are shown in Table 2. In both the flat and corner portions, similar material properties has been assumed in the FE analysis and hence, strength enhancement in the corner regions by cold forming process has been neglected following Patton and Singh [16,40]. The flat material property in Fig. 3 is then used as input in the ABAQUS model by pl ) using the converting into true stress (σtrue) and true plastic strain (εtrue following Eqs. (2) and (3).

2.4. Geometric imperfections In order to seed geometric imperfections into the FE models, both local and geometric buckling modes shapes were extracted through linear elastic eigen buckling analysis, using the Lanczos method [34]. The load-deformation response was captured by using the modified Riks method [41,42]. The lowest local and global mode shapes were used to perturb the model geometry, by scaling with local imperfection amplitude of t/100 [13, 26, 32], where t is the thickness; and global or flexural imperfection amplitude of l/1500 [13,21,40], where l is the height of the column, respectively.

σtrue = σnom(1 + εnom )

ε pl true = ln(1 + εnom ) −

Gardner and Ashraf [43]'s model (modified version of original Ramberg–Osgood [44], Rasmussen [45]) has been used to define the stress strain curve of LDSS material used in this work. The key material parameters (Young's modulus, E, 0.2% proof stress, σ0.2, 1% proof stress, σ1.0, strain hardening coefficients, n and n'0.2,1.0) used in the Gardner and Ashraf [43]'s model are shown in Table 1. It may be noted that, as per EN 10088-4 [46], the specified minimum mechanical properties of LDSS Grade EN 1.4162 are 530 MPa for 0.2% proof stress (σ0.2) and ultimate stress (σu) value of 700–900 MPa. The value of Poisson's ratio is considered to be 0.3. The stress strain curve proposed

80×80×4 60×60×3

197200 206400

σ0.2 (MPa)

657 711

σ1.0 (MPa)

770 845

Compound R-O coefficients n

n'0.2,1.0

4.7 5.0

2.6 2.7

(3)

Table 2 Slender column dimensions [13].

Table 1 Compressive LDSS flat material properties [13]. E (MPa)

σtrue Eo

where σnom and εnom are engineering stress and strain respectively. Non-linear FE analyses was then performed using Riks method [41,42], after incorporating both local and global imperfections. The validation is presented in the form of load (P) vs lateral mid-length deflection (δlat) in Fig. 4. It can be seen that both the ultimate and post-ultimate curve profiles are well matched, thus indicating that the current FE modeling approach can provide reasonably accurate results for the modeling of LDSS hollow columns under axial compression. Hence, this

2.5. Material modeling

Cross-section

(2)

Specimen

D (mm) B (mm) t (mm) ri (mm)

Buckling length , lcr (mm)

80×80×4–1200 80×80×4–2000 60×60×3–2000 60×60×3–1600 60×60×3–1200 60×60×3–800

79.3 79.6 60 59.6 60 60

1199.5 1999.0 1999.0 1599.0 1199.0 799.0

79.6 79.5 60 60 60 60

3.72 3.80 3.13 3.15 3.13 3.13

3.8 3.4 2.7 2.4 2.4 2.4

lcr=Buckling length, D=Depth, B=Width, t=thickness, ri=internal corner radius.

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(t=10–20 mm). In all, a total of ~120 FE flat oval hollow slender column models were analysed. The labelling convention followed the pattern flat length (lf), distance between flat lengths (w), radius of curve element (r), thickness (t) and height (l). For example, lf50w50r25t2l1000 would represent a column with lf=50 mm, w=50 mm, r=25 mm, t=2 mm and l=1000 mm. Based on the FE analyses, structural performance of the columns under axial compression were assessed, with attention towards, load capacity and deformation modes. Special observations are also made on the failure modes or mechanism both at ultimate (i.e. δu) and post-ultimate (e.g. ~1.5δu) deformations, for both the slender and stocky cross sections. Later, the FE results were compared with the predictions made by stainless steel design codes such as EN 1993-1-4 [35] and AS/NZS 4673 [36], to check their applicability for LDSS flat oval hollow columns. Further, the FE results are also compared with proposed modified AS/NZS 4673 and DSM specifications made by Huang and Young [9] (based on their experimental studies on LDSS hollow square and rectangular sections). Before, proceeding with the presentation of results and discussions (Section 5), the codal and modified codal, and design method considerations are presented in the following sub-sections. 3.1. Codal and proposed modified design considerations Australian/New Zealand Standard (AS/NZS 4673) [36], European (EN 1993-1-4) Code [35] American Society for Civil Engineers (ASCE-802) Standard [50], Automotive Steel Design Manual (ASDM) [51] etc. (hereafter referred to as ‘codal’) have provisions for the design of cold formed stainless steel structural members. It may be noted that, these codes do not explicitly provide guidelines for the structural design of hollow steel members having flat oval cross sections. Hence, as mentioned above, modifications have been suggested for the codal provisions, by Huang and Young [9]. In this section, the EN 1993-1-4 [35], AS/NZS 4673 [36], DSM [37], modified AS/NZS 4673, modified DSM procedures by Huang and Young [9] for calculating the design strengths are briefly described below.

Fig. 3. Experimental stress-strain curve of LDSS material Grade EN 1.4162 for different samples of (a) 80×80×4 and (b) 60×60×3 (Theofanous and Gardner [13]).

FE- (80x80x4-1200)

3.1.1. EN 1993-1-4 [35] The column design strength of EN 1993-1-4 [35] i.e. (PEN) as per Clause 5.4.1 of EN 1993-1-4 [35] are given in Eq. (4)

Exp (80x80x4-2000)

PEN = χfy A

Exp (80x80x4-1200)

800 80x80x4-1200

FE- (80x80x4-2000)

600

where the buckling reduction factor, χ (to account for flexural buckling) is defined as;

Exp (60x60x3-800) P (kN)

60x60x3-800

FE-(60x60x3-800)

400

Exp (60x60x3-1200)

80x80x4-2000

10

20

30

Exp (60x60x3-2000)

40

50

≤ 1.0; where φ = 0.5(1 + α(λ − λ 0 ) + λ 2 ) (5)

φ is capacity (strength reduction) factor. The non-dimensional slenderness ratio, λ , is given by Eq. (6); λ=

FE- (60x60x3-2000) 0

0.5

φ + [φ − λ 2 ]

FE- (60x60x3-1600) 60x60x3-2000

0

2

Exp (60x60x3-1600)

200

1

χ=

FE- (60x60x3-1200)

60x60x3-1200

60x60x3-1600

(4)

60

Afy Ncr

(6)

Fig. 4. Validation of P (load) vs δlat (lateral displacement) for SHC 80×80×4 (l=1200 and 2000 mm) and SHC 60×60×3 (l=800, 1200, 1600, 2000 mm) (Theofanous and Gardner [13]).

where, A is the cross-sectional area. As per EN 1993-1-4 [35], A is taken as the gross cross-sectional area (Ag) for Class 1 and 2 sections. Ncr is the elastic critical force for the relevant buckling mode based on gross cross sectional properties given by Eq. (7).

FE modeling approach is adopted for the current study.

Ncr =

3. Parametric study of LDSS flat oval hollow slender column

where, E, I and le are the Young's modulus, moment of inertia and effective height of the column taking into consideration of the boundary conditions respectively. The values of α (imperfection factor) and λο (limiting slenderness) considered in this study are 0.49 and 0.40, respectively (see Table 5.3 of the EN 1993-1-4 [35]).

įlat (mm)

As mentioned in Section 2, FE analyses of LDSS flat oval hollow sender columns were carried out by varying the geometrical parameters like flat element length (lf=50–150 mm), radius of the curve element (r=25–100 mm), distance between the flat elements (w=50–200 mm), length/height of the columns (l=1000–3500 mm), and thickness 50

π 2EI le 2

(7)

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3.1.2. AS/NZS 4673 [36] The column design strength based on AS/NZS 4673 [36] i.e. (PAS/NZS) is given in Eq. (9)

PAS / NZS = Afn

(9)

where fn is the stress in the column member i.e. the least of the flexural, torsional and flexural-torsional buckling stress and is given by Eq. (10)

fn =

fy ϕ+

ϕ2 − λ f 2

≤ fy (10)

1 (1 + η + λ f 2 ) 2

(11)

⎛ jl ⎞ fy λf = ⎜ ⎟ 2 ⎝r ⎠ π E

where j, r, l are the effective length factor, radius of gyration and unbraced total length of column respectively, and (13)

λl

Existing Coefficients for EN 1.4462 (Duplex) Proposed Coefficients for EN 1.4162 (Lean duplex)

1.16 0.86

0.13 0.14

0.65 0.67

0.42 0.45

⎧ Pne for λl ≤ 0.769 ⎪ ⎪ ⎛ Pcrl ⎞0.4 ⎤⎛ Pcrl ⎞0.4 Pnl = ⎨ ⎡⎢ ⎪ 1 − 0.16⎜ Pne ⎟ ⎥⎜ Pne ⎟ Pne for λl > 0.769 ⎪ ⎢⎣ ⎝ ⎠ ⎥⎦⎝ ⎠ ⎩

(19)

In order to assess the applicability of the design codes such as EN 1993-1-4 [35], AS/NZS 4673 [36], DSM [37]; and modified AS/NZS 4673, modified DSM [9] for flat oval slender columns, reliability analyses have been carried out considering their respective resistance factor, load combination of the codes and method with the prediction and current FE results. The reliability analyses were carried out based on the procedures outlined in the ASCE Specification commentary [50] for cold formed steel structures. In the estimation of reliability index, the resistance factor (Ф) value of 0.85, 0.91, are considered for the AS/ NZS [36] and EN 1993-1-4 [35] respectively. For section which are not pre-qualified, it is suggested by the NAS [38] that Ф=0.80 may be taken for DSM. The load combinations viz., 1.25DL+1.5LL and 1.35DL +1.5LL (where LL and DL are the live and dead loads) are considered as suggested by AS/NZS 4673 [36] and EN 1993-1-4 [35]. NAS [38] recommends the ratio of DL to LL be taken as 0.20. The load combination for the DSM is taken as 1.2DL+1.6LL i.e. similar to that of ASCE [50] (Zhu and Young [20]). As per ASCE specification [50], the values of VM, VF, Mm, and Fm (i.e. statistical parameters values of mean values, coefficients of variation (COV) of fabrication and material) are taken as 0.10, 0.05,1.10, and 1.0. Correction factor (Cp) according to Eqs. (F1.1)–(F1.3) of NAS specification [54] has also been taken into account for the effect of the number of data. A lower bound value of 2.5 for the reliability index (βo) has been considered for a reliable design, as per the ASCE norm.

Table 3 Existing and proposed new coefficients of buckling stress in AS/NZS standard (Huang and Young [9]). λo

(18)

4. Reliability analysis

(14)

b

2 ⎧ (0.87λc )Py for λc ≤ 1 ⎪ ⎪ pne = ⎨ ⎛ ⎞ ⎪ ⎜ 0.877 ⎪ ⎝ λc2 ⎟⎠Py for λc ≤ 1 ⎩

A comparison of DSM (Eqs. (15) and (16)) and modified DSM (Eqs. (18) and (19)), shows that the difference between the DSM and modified DSM are in the values of coefficients of Py and Pne; and on the limits of λl and λc.

3.1.4. Direct strength method Cold formed steel members can experience local, global and distortional buckling modes, and a new design approach (popularly known as Direct Strength Method, DSM) was put forward by Schafer and Pekoz [52] for their design. The attractive part of the DSM is that the estimation of effective width is not required, rather elastic critical loads are computed for all the buckling modes such as local, global and distortional buckling modes. Further, full cross-section is used to compute the slenderness ratio, instead of considering the weakest/ slender constituent plate elements. Detailed DSM design steps are provided in the Appendix 1 of NAS [38]. The critical elastic loads for complex sections are generally computed using finite element method [34] or finite strip method [53]. In this study, for the flat oval section under consideration, distortional modes are neglected as the section is closed (see e.g. Zhu and Young [21]). Eq. (14)–(16) shows the relevant DSM equations for cold formed steel [38,54]. The DSM nominal axial strength (PDSM) is then given by Eq. (14)

a

(17)

where,

3.1.3. Proposed modified AS/NZS 4673 (Huang and Young [9]) As mentioned before, LDSS material particularly EN 1.4162 is not covered in the current AS/NZS 4673 [36]. Thus, based on their study, Huang and Young [9] on LDSS square and rectangular hollow members, new bucking stress coefficient values i.e. 0.86, 0.14, 0.67 and 0.45 for α, β, λo and λl, respectively, were suggested for members made of LDSS in Table 3. With the new buckling coefficient values, modified AS/NZS 4673 [9] design strength (PAS/NZS-m) are estimated for LDSS columns.

Coefficients

(16)

pDSM − m = min(Pne, Pnl )

In the above expression (Eq. (13)), α, β, λo and λl denote the buckling stress coefficients. As the current AS/NZS 4673 [36] does not provide the values of these coefficients for LDSS (EN 1.4162) material, the values corresponding to S31803 (EN 1.4462, duplex) i.e. 1.16, 0.13, 0.65 and 0.42 for α, β, λo and λl, respectively (see Table 5 of AS/NZS 4673 [36]), have been used in deriving the column strength.

pDSM = min(Pne, Pnl )

⎧ Pne for λl ≤ 0.776 ⎪ ⎪⎡ 0.4 ⎤ 0.4 ⎛ Pcrl ⎞ ⎛ Pcrl ⎞ Pnl = ⎨ ⎢ ⎪ 1 − 0.15⎜ Pne ⎟ ⎥⎜ Pne ⎟ Pne for λl > 0.776 ⎪ ⎢⎣ ⎝ ⎠ ⎥⎦⎝ ⎠ ⎩

3.1.5. Proposed modified DSM (Huang and Young [9]) A modification to the original DSM was reported by Huang and Young [9] based on their studies on square and rectangular LDSS hollow columns. The modified nominal axial strength, PDSM-m (for LDSS columns) is given in Eq. (17).

(12)

η = α((λ f − λ1) β − λ 0 )

(15)

Here, Py=fyA; λc = Py / Pcre ; λl = Pne / Pcrl . Critical elastic buckling load in flexural buckling, Pcre=π2EA/(le/ry)2, where Pcrl, ry, are critical elastic local buckling load, radius of gyration of gross cross-section about the minor axis of buckling. Pnl is the nominal axial strength for local buckling as well as interaction of local and overall/global buckling respectively.

The expression for ϕ (capacity factor) and λ f (slenderness ratio) are given in Eqs. (11) and (12).

ϕ=

2 ⎧ (0.658λc )Py for λc ≤ 1.5 ⎪ ⎪ pne = ⎨ ⎛ ⎞ ⎪ ⎜ 0.877 ⎪ ⎝ λc2 ⎟⎠Py for λc ≤ 1.5 ⎩

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5. Results and discussion The non-linear FE analyses results of the flat-oval LDSS hollow slender columns under axial compression, for the parameter range mentioned in Section 3 are presented in the following sub-sections. The results are presented in the form of different deformation or failure mechanisms, followed by the effect of cross-sectional parameters like r, lf, w, and l, on the load capacity (Pu). Afterwards, the FE results are compared with predictions from EN 1993-1-4 [35], AS/NZS 4673 [36], DSM [37,52]; and modified AS/NZS 4673 [9], modified DSM [9], to check their efficiency for flat-oval LDSS hollow slender columns. 5.1. Failure mechanisms For the stocky cross-section sections (i.e. Class 1 and 2) considered in this study, based on the FE analyses of ~120 slender models, three failure or deformation mechanisms have been identified viz., 1) Yield to yield with local bucking (Mechanism ST1), 2) Yield to yield with flexural buckling (Mechanism ST2), and 3) Flexural to flexural buckling (Mechanism ST3). These classifications are based on observations of von-Mises contour superimposed in deformed shapes, at ultimate load and post ultimate load i.e. at δu and 1.5 δu. 5.1.1. Yield to yield with local bucking (Mechanism ST1) Fig. 5 shows the typical variation of P/Pcr and P/Py (where Py=Agfy is the gross yield load) with δ, for yield to yield with local buckling (or Mechanism ST1). The plots correspond to specimen lf50w150r75t15l1000. From Fig. 5b and c, it can be seen that Pu/ Pcr=~0.08 and Pu/Py=~1.32. The lower value of Pu/Pcr may be related to the high value of elastic buckling load associated with relatively short column with stocky cross-section (l/lf=20 and λ =0.17). A high value of Pu/Py or Pu/Py > 1.0, indicates that the column is able to withstand load more than the gross yield, the additional strength may then be related to the strain hardening of high strength LDSS steel. Due to this high load capacity (beyond the gross yield load), a gradual drop (Pu/Py=1 at δ/l =~7%) and rounded load deformation curve around the peak can be seen. At δu, almost the column has yielded without any signs of bucking (inset Fig. 5P1), and however, at 1.5δu, three outward local buckling (with half buckling wavelength ~11% of column length) can be observed both at mid-

Fig. 6. Typical mechanism ST2 (yielding to flexural buckling) failure mode: a) von-Misses contour plot of Q1 (at δu) and Q2 (at 1.5δu); b) variation of P/Pcr with δ (axial deformation), and c) variation of P/Py with δ (lf50w150r75t15l2000).

length and the two end spots. It can be noticed that even at 1.5δu, most region are still yielded with small region in the flat elements near the buckled portion being distressed. 5.1.2. Yield to yield with flexural buckling (Mechanism ST2) Typical plots of variation of P/Pcr and P/Py with δ, and von-Mises contour plots are shown in Fig. 6, for yield to yield with flexural bucking mechanism (Mechanism ST2). The plots correspond to specimen lf50w150r75t15l2000. It can be seen from Fig. 6, that P/Pcr=0.17 and P/Py=~1.18. As compared to the previous Mechanism ST1, the cross-sectional parameters are same, except for the increase (~100%) in l to 2000 from 1000 mm, with λ =~0.35. The effect of this can be seen in increase (~112%) of P/Pcr and decrease (~10%) in P/Py as compared to the specimen considered for Mechanism ST1 (see Fig. 5). At the ultimate load (or δu), the specimen is almost yielded (with no instance seen for local buckling), whereas, at 1.5δu, flexural buckling can observed with most region still being yielded (distressed at the ends for ~29% column can be seen at the flat element side). The postultimate load-deformation curve (see Fig. 6c) appears to be relatively sharper, with P/Py=1.0 occurring at ~1.3% of the column length, this again is relatively shorter as compared to the specimen discussed in the previous section (Fig. 5). 5.1.3. Flexural to flexural buckling (Mechanism ST3) Fig. 7 shows the plots (variation of P/Pcr and P/Py with δ, and vonMises contour plot) illustrating the flexural to flexural buckling mechanism (or Mechanism ST3). The behaviour shown in Fig. 7 relates with the specimen lf50w50r75t15l2000, resulting in much larger λ value (~1.1), as compared to the specimens discussed previously. As compared to the specimens with the above mentioned ST1 and ST2 mechanisms, in this specimen, the value of Pu/Pcr is much larger (~1.0) however, the value of Pu/Py has dropped to 0.67, suggesting that global behaviour is in the elastic range (the pre-Pu load-deformation curve is again near linear). At Pu, the curvature of the loaddeformation curve is now relatively sharper with a faster drop in postPu load. In Fig. 7(a), it is seen that at δu, there is an initiation of flexural buckling with compression yielded regions at the mid-length and the two end supports (characteristic of fixed ended boundary conditions).

Fig. 5. Typical mechanism ST1 (yielding to yielding buckling) failure mode: a) vonMisses contour plot of P1 (at δu) and P2 (at 1.5δu); b) variation of P/Pcr with δ (axial deformation) and c) variation of P/Py with δ (lf50w150r75t15l1000).

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capacity of slender columns with stocky sections is presented in Fig. 9 (lf=50 mm, w=50 mm, t=15 mm, r=25–100 mm, for l=1000– 3500 mm. It can be seen that semi-circular curve section (r/w=0.5) showed the highest load capacity, with Pu remaining almost flat for r/ w≥1.0; with the drop in Pu when r/w is increased from 0.5 to 1, is ~18–20% for all the values of l considered. It may be noted that as r tends to zero (or r/w→0, in this case), the section would approach to flat plate of thickness 2t and width w, as also noted by Ruiz-Teran and Gardner [55] for the elastic analysis of elliptical sections. The results corresponding to such flat plates are also plotted in Fig. 9(a). Clearly, Pu values corresponding to r=0.0 (or r/w=0) are relatively much lower, owing to the absence of curve elements, when compared to flat oval hollow sections. Fig. 9(c) shows the variation of Pu/Agfy with r/w, It can be seen that Pu/Agfy > 1.0 for l=1000 mm, whereas for l≥2000 mm, Pu/Agfy < 1.0. Pu/Agfy > 1.0 is due to strain hardening. It is also observed that the data points corresponding to Pu/Agfy < 1.0, falls under Mechanism ST3 (from l=2000–3500 mm), whilst the data points belonging to l=1000 mm, corresponds to Mechanism ST2. It may be observed that the rise in Pu/Agfy may be related to the decrease in Ag as r/w increases in Fig. 9(b) (or as r become flatter), although the drop in Ag for 1≤r/w is very marginal.

1

5.2.2. Effect of flat length (lf) Fig. 10 shows the effect of flat length element (lf) on Pu and Pu/Agfy with increasing columns height from 1000 to 3500 mm (w=50 mm, r=25 mm, t=15 mm, lf=50–150 mm). From Fig. 10(a), it appears that there is near linear increase in Pu with lf; the average increase is found to be ~94–110% for 200% increase in lf (from lf=50 mm). This is also consistent with the linear increase in Ag with increase in lf in Fig. 10(b). Further, as, lf→0, the section approaches circular hollow section, similar to the argument put forward by Ruiz-Teran and Gardner [55] wherein the behaviour of elliptical section would tend towards circular as the aspect ratio→1. In Fig. 10(a), results for corresponding circular sections (by making lf=0) are presented. The reduction in Pu values for the circular sections (or lf=0) are consistent with the absence of stiffening contribution of the two flat elements. The variation of Pu/Agfy with lf in Fig. 10(c), is seen to be near horizontal for all the values of h, although the shortest height column (h=1000 mm) resulted in Pu/Agfy > 1.0, whereas for l≥2000, Pu/ Agfy < 1.0. Further, data points corresponding to Pu/Agfy < 1.0, resulted in ST3 mechanism, whilst those corresponding to Pu/Agfy > 1, resulted in ST2 mechanism.

Fig. 7. Typical mechanism ST3 (flexural to flexural buckling) failure mode: a) von-Misses contour plot of R1 (at δu) and R2 (at 1.5 δu); b) variation of P/Pcr with δ (axial deformation), and c) variation of P/Py with δ (lf50w50r75t15l2000).

At 1.5δu, enhanced flexural type continue to happen with the compression yielded region covering ~28% of the column length (inset Fig. 7R2); the yielded region at the ends are also increased (~12.5% of column length), however, it now shifted towards the tension side. Variation of Pu/Agfy with λ , for all the 120 data points are shown in Fig. 8, with data points corresponding to the above mentioned three Mechanisms i.e. ST1, ST2 and ST3 being marked. The data points corresponding to the three typical specimens considered for illustrating the three failure/deformation mechanisms are also highlighted. From the Fig. 8, roughly, it is now possible to demarcate the limits of the three failure mechanisms e.g. λ < 0.32; 0.32≤λ ≤1.0; and λ > 1.0 corresponds to ST1, ST2 and ST3 mechanisms respectively. It may also be noted that both ST1 and ST2 (or combinely ST) corresponds to yield initiated failure mechanism at Pu, whilst ST3, corresponds to flexure buckling initiated mechanism.

5.2.3. Effect of distance between flat elements (w) Effect of distance between flat elements (w) is presented in the form of variation of Pu/Agfy with w, in Fig. 11, (lf=50 mm, w/r=2, t=15 mm, w=50–200 mm), for l=1000–3500 mm. In Fig. 11, it can be seen that Pu/Agfy≥1.0 and remains near flat, for w≥100 mm. However, for w < 100 mm and l≥2000 mm, Pu/Agfy < 1.0. At lower values of w, e.g. w close to 50 mm, the columns become relatively slender so as to initiate the failure by flexure at Pu, especially for l≥2000 mm, as can be seen from the inset Fig. 11L3 and L4. These behaviour is in agreement with the association of Mechanism ST3 (i.e. flexure initiated deformation) for w≤100 mm and l≥2000 mm. For w≥100 mm and l=2000–3500 mm, Pu/Agfy > 1, due to the failure Mechanism ST2 (i.e. yield to yield with flexural buckling).

5.2. Variation of buckling loads 5.2.1. Effect of curvature radius (r) The effect of curvature radius (r) of the curve elements on the load 1.5

ST1

0..32

1.0

ST2

Mechanism-STI

Pu/Agfy

1.0

Mechanism-ST2

ST3

Mechanism-ST3

0.5

0.0

0.32 0

0.5

5.2.4. Effect of thickness (t) Fig. 12 shows the effect of thickness (t) on Pu and Pu/Agfy for lf=50 mm, w=50 mm, and r=100 mm, for variation of t from 10 to 20 mm. From Fig. 12(a), there appears to show a near linear increase in Pu, with the rate of increase of t for all height of columns but decreases with increase in l. The increase in Pu with increased in thickness correlate well with the linear increase in gross-area of cross-section as shown in Fig. 12(b). For 200% increase in t (from 10 mm), the increase in Pu for l=1000

1.0 1 _

1.5

2

2.5

Fig. 8. Variation of Pu/Agfy with λ showing Mechanisms ST1, ST2 and ST3 regions with t=10–20 mm.

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Fig. 9. Variation of (a) Pu vs r/w, (b) Ag vs r/w and (c) Pu/Agfy vs r/w (lf=50 mm, w=50 mm, t=15 mm, r=25–100 mm).

5.2.5. Effect of height (l) Effect of LDSS flat oval hollow column height (l) on Pu is presented in Fig. 13(a–c); for t=15 mm. Fig. 13(a) shows the variation of Pu with l for lf=50–150 mm (w=50 mm, r=25 mm). It can be seen from Fig. 13(a) that Pu decreases with increasing l in a near-non-linear trend; the decrease being nearly constant ~ 77% for the range of lf considered. However the values of Pu are found to be higher for longer lf. The effect of l on Pu, for r=25–100 mm (lf=50 mm, w=50 mm) is shown in Fig. 13(b). Although, a near-non-linear drop in Pu with l (~77% and ~74% for r=25 mm and 50–100 mm respectively) is observed in Fig. 13(b), it is seen that Pu values corresponding to r=25 mm (i.e. when the curve element is semi-circular or r/w=0.5) are

and 3500 mm are ~88% and ~46% respectively. In Fig. 12(c), it is seen that for Pu/Agfy < 1.0 which occurred for l≥2000 mm, the curves are almost flat, however for Pu/Agfy > 1.0 (associated for l=1000 mm), the variation showed a slight increase. It may be related to the associated failure mechanisms: l=1000 mm, showing ST2 mechanism whilst l≥2000 mm, showed ST3 mechanism. For the columns with ST2 mechanism, the increasing trend in Pu/Agfy, the role of strain hardening may have become load enhancing role with increasing Ag. However, for the longer sections, the decreasing effect of Pu with increasing l, may have been neutralised by the opposite effect (increase of Pu) of increasing Ag in Fig. 12(b), and hence have resulted in near flat behaviour of Pu/Agfy.

Fig. 10. Variation of (a) Pu vs lf, (b) Ae vs lf and (c) Pu/Agfy vs lf (w=50 mm, r=25 mm, t=15 mm, lf=50–150 mm).

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in lf and r; however, it is quite sensitive (~8–77%) to variation in w. This may be because, increasing w (maintaining r/w at 0.5) can relatively lower the member slenderness as compared to increasing lf or r. Variation of Pu/Agfy with λ for changes in lf, r and w are also plotted for the same date set considered in Fig. 13(d–f). It is interesting to note the normalised plots are able to capture well the effects of r and lf (see the near convergence of the plots in Fig. 13(d) and (e)). However, such well behaved plots are not seen when w is varied (see Fig. 13(f)). This is essentially because, for the case of w=50 mm, most of the slender (λ > ~1) specimens failed by flexure-flexure or ST3 mechanism (see Table 4), whilst the other specimens failed by either mechanism ST1 or ST2 (note Pu/Agfy > 1.0). 5.3. Comparison of numerical results with design strengths Comparison of the prediction from design codes (EN-1993-1-4 [35], AS/NZS 4673 [36]) and method (DSM [37]) and their proposed modified versions (Huang and Young [9]) with the results from FE analyses are shown in Figs. 14–16, for stocky LDSS flat oval hollow columns. Unlike the thin section, full cross-section is taken as effective section in case of thick section and so the design curve of EN 1993-1-4 has been shown in Fig. 14(a). The comparisons are shown in the form of variation of Pu/Agfy vs λ for EN 1993-1-4 [35], Pu/Agfy vs λf for AS/NZS 4673 [36], Pu/Py vs λc for DSM [37]. The FE results are based on 120 models with slenderness (λ or λf or λc) ratio in ranges of ~0.12–2.25. The ratio of Pu from FE analyses (i.e. Pu) and those of design predictions are shown in Figs. 14(b), 15(b) and 16(b) to show the scattering of the compared results. It can be seen that EN-1993-1-4 [35], AS/NZS 4673 [36], and the modified version of AS/NZS [9] are conservative. However, although DSM and modified DSM is able to provide conservative values of Pu, it is seen that some of their predictions are found to be under conservative (specially for 0.63 < λ < 1.2), as indicated by Pu/PDSM < 1.0 in Fig. 16(b). The results of the reliability analyses are presented in Table 4, based on the reliability analyses and considering the

Fig. 11. Variation of Pu/Agfy vs w (lf=50 mm, w/r=2, t=15 mm, w=50–200 mm).

the highest. For r=50–100 mm, the Pu results are found to be similar. Variation of Pu with l for w=50–200 mm (lf=50, r/w=0.5 or semicircular curve section) is shown in Fig. 13(c). It can be observed that a near-linear decrease in Pu can be seen in Fig. 13(c), with the decrease being lowest (~8%) for w=200 mm and highest (~77%) for w=50 mm. Thus, it can be observed that, although the decrease in Pu with l is quite expected (due to increase in global/member slenderness), the rate of decrease is relatively less sensitive (~74–77%) to variations

Fig. 12. Variation of (a) Pu vs t, (b) Ag vs t and (c) Pu/Agfy vs t (lf=50 mm, w=50 mm, r=100 mm).

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Fig. 13. Pu/Agfy vs h (a–c) and Pu/Afy vs λ (d–f) at t=15 mm for w=50 mm, r=25 mm, lf=50–150 mm (a,d); lf=50 mm, w=50 mm, r=25–100 mm (b,e); lf=50 mm, r/w=0.2, w=50–200 mm (c,f).

with the fitted Euler column formula have been shown in Fig. 14(a) for EN 1993-1-4 [35] and in Fig. 15(a) for AS/NZS 4673 [36]. Euler formula is found to be applicable for prediction of strength of column at higher slenderness values λ or λ≥1.5 but for lower slenderness values, the rate of deviation of Euler curve from FE values increases with the decrease in slenderness of section mainly due to the strain hardening effect of the column.

minimum requirement value of the reliability index (βo) as 2.5 (ASCE specification [50]), it can be inferred that all the compared design codes, method and their modified versions are reliable and applicable, with βo value of 3.28, 3.52, 3.90 for EN-1993-1-4 [35], AS/NZS 4673 [36] and DSM [37]; and 3.38, 3.50 for modified AS/NZS and modified DSM (Huang and Young [9]) respectively. However, in case the reliability resistance factor (Ø) and load combination has been made same with values of 0.85 and 1.25DL+1.5LL, the reliability index values for the current EN 1993-1-4 [35], AS/NZS 4673 [36] and DSM [37] are 3.49, 3.52, 3.46; while the proposed modified AS/NZS 4673 and DSM (Huang and Young [9]) gives the values of 3.38 and 3.07 respectively. It has been noticed that βo values for the modified versions are found to be lower and hence, their use are expected to be more economical as well as reliable. The prediction of strength of column

6. Conclusions A parametric investigation on the structural behaviour of fixed ended stocky LDSS flat oval slender columns under axial compression has been presented using finite element analyses, considering variations on element radius, flat element length, flat element separation distance, 56

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Table 4 Comparison of FE strength with design strengths for stocky sections (t=10–20 mm). Specimen

P/Py

Failure mode at δu

Failure mode at 1.5δu

PFE PEN

PFE PAS / NZS

PFE P* AS / NZS

lf50w50r25t10l1000 lf50w50r25t10l2000 lf50w50r25t10l2500 lf50w50r25t10l3500 lf75w50r25t10l1000 lf75w50r25t10l2000 l75w50r25t10l2500 lf75w50r25t10l3500 lf100w50r25t10l1000 lf100w50r25t10l2000 lf100w50r25t10l2500 lf100w50r25t10l3500 lf150w50r25t10l1000 lf150w50r25t10l2000 lf150w50r25t10l2500 lf150w50r25t10l3500 lf50w50r50t10l1000 lf50w50r50t10l2000 lf50w50r50t10l2500 lf50w50r50t10l3500 lf50w50r75t10l1000 lf50w50r75t10l2000 lf50w50r75t10l2500 lf50w50r75t10l3500 lf50w50r100t10l1000 lf50w50r100t10l2000 lf50w50r100t10l2500 lf50w50r100t10l3500 lf50w100r50t10l1000 lf50w100r50t10l2000 lf50w100r50t10l2500 lf50w100r50t10l3500 lf50w150r75t10l1000 lf50w150r75t10l2000 lf50w150r75t10l2500 lf50w150r75t10l3500 lf50w200r100t10l1000 lf50w200r100t10l2000 lf50w200r100t10l2500 lf50w200r100t10l3500 lf50w50r25t15l1000 lf50w50r25t15l2000 lf50w50r25t15l2500 lf50w50r25t15l3500 lf75w50r25t15l1000 lf75w50r25t15l2000 lf75w50r25t15l2500 lf75w50r25t15l3500 lf100w50r25t15l1000 lf100w50r25t15l2000 lf100w50r25t15l2500 lf100w50r25t15l3500 lf150w50r25t15l1000 lf150w50r25t15l2000 lf150w50r25t15l2500 lf150w50r25t15l3500 lf50w50r50t15l1000 lf50w50r50t15l2000 lf50w50r50t15l2500 lf50w50r50t15l3500 lf50w50r75t15l1000 lf50w50r75t15l2000 lf50w50r75t15l2500 lf50w50r75t15l3500 lf50w50r100t15l1000 lf50w50r100t15l2000 lf50w50r100t15l2500 lf50w50r100t15l3500 lf50w100r50t15l1000 lf50w100r50t15l2000 lf50w100r50t15l2500 lf50w100r50t15l3500 lf50w150r75t15l1000

1.16 0.83 0.67 0.30 1.13 0.83 0.52 0.32 1.09 0.83 0.69 0.32 1.04 0.83 0.68 0.34 1.18 0.80 0.62 0.30 1.00 0.59 0.46 0.25 1.20 0.72 0.56 0.31 1.31 1.16 1.12 1.00 1.28 1.16 1.13 1.03 1.25 1.25 1.24 1.07 1.14 0.62 0.47 0.26 1.19 0.61 0.47 0.27 1.11 0.61 0.47 0.27 1.14 0.60 0.49 0.28 1.28 0.69 0.54 0.28 1.32 0.72 0.55 0.29 1.33 0.75 0.56 0.30 1.32 1.25 1.16 1.06 1.33

Y F F F Y F F F Y F F F Y F F F Y F F F Y F F F Y F F F Y Y Y Y Y Y Y Y Y Y Y Y Y F F F Y F F F Y F F F Y F F F Y F F F Y F F F Y F F F Y Y Y Y Y

F F F F F F F F F F F F F F F F F F F F F F F F F F F F Y F F F Y F F F Y Y Y F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Y

1.27 1.53 1.69 1.32 1.22 1.47 1.24 1.31 1.16 1.78 1.59 1.30 1.09 1.39 1.52 1.28 1.30 1.51 1.61 1.36 1.33 1.36 1.45 1.40 1.31 1.37 1.47 1.43 1.21 1.25 1.32 1.52 1.13 1.12 1.15 1.17 1.08 1.15 1.19 1.10 1.29 1.28 1.36 1.32 1.35 1.29 1.40 1.40 1.26 1.29 1.40 1.43 1.29 1.28 1.45 1.46 1.43 1.38 1.50 1.36 1.45 1.41 1.47 1.39 1.46 1.42 1.47 1.37 1.22 1.37 1.40 1.69 1.18

1.35 1.55 1.63 1.22 1.28 1.50 1.20 1.22 1.21 1.48 1.55 1.21 1.13 1.44 1.49 1.20 1.39 1.52 1.55 1.26 1.43 1.37 1.40 1.30 1.40 1.38 1.41 1.33 1.31 1.31 1.45 1.60 1.28 1.16 1.13 1.28 1.25 1.25 1.24 1.07 1.40 1.27 1.29 1.22 1.47 1.28 1.33 1.30 1.37 1.47 1.33 1.32 1.40 1.27 1.37 1.35 1.54 1.38 1.43 1.26 1.56 1.41 1.41 1.29 1.56 1.43 1.41 1.27 1.32 1.46 1.54 1.76 1.33

1.21 1.40 1.51 1.17 1.15 1.35 1.11 1.16 1.09 1.33 1.43 1.15 1.04 1.29 1.36 1.14 1.25 1.38 1.44 1.21 1.29 1.24 1.29 1.25 1.26 1.25 1.31 1.27 1.31 1.17 1.30 1.44 1.28 1.16 1.13 1.15 1.25 1.25 1.24 1.07 1.26 1.16 1.20 1.17 1.33 1.17 1.25 1.25 1.23 1.34 1.24 1.27 1.27 1.16 1.28 1.30 1.39 1.25 1.33 1.21 1.41 1.28 1.31 1.24 1.41 1.29 1.31 1.22 1.32 1.31 1.38 1.58 1.33

57

PFE PDSM

PFE * PDSM

1.31 1.21 1.33 1.07 1.41 1.35 1.19 1.19 1.27 1.18 1.30 1.00 1.03 0.97 1.17 1.17 1.21 1.13 1.28 0.97 1.33 1.24 1.15 1.15 1.15 1.07 1.24 0.95 1.28 1.17 1.13 1.13 1.34 1.23 1.31 1.07 1.34 1.30 1.23 1.23 1.38 1.26 1.18 0.90 1.21 0.97 1.27 1.05 1.35 1.25 1.19 0.98 1.22 1.18 1.29 1.29 1.35 1.32 1.30 1.20 1.33 1.19 1.40 1.12 1.29 1.28 1.22 1.18 1.22 1.16 1.19 1.08 1.26 1.25 1.28 1.26 1.30 1.26 1.17 1.10 1.32 1.20 1.09 0.95 1.13 1.13 1.21 1.21 1.38 1.25 1.09 0.97 1.18 1.17 1.29 1.29 1.28 1.17 1.26 1.12 1.17 1.17 1.32 1.32 1.32 1.19 1.08 0.96 1.21 1.21 1.35 1.35 1.47 1.34 1.18 1.01 1.24 1.23 1.24 1.24 1.50 1.37 1.21 1.02 1.22 1.20 1.26 1.26 1.51 1.39 1.23 1.01 1.22 1.18 1.24 1.24 1.36 1.33 1.41 1.31 1.40 1.23 1.53 1.20 1.35 1.34 (continued on next page)

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Table 4 (continued) Specimen

P/Py

Failure mode at δu

Failure mode at 1.5δu

PFE PEN

PFE PAS / NZS

PFE P* AS / NZS

PFE PDSM

PFE * PDSM

lf50w150r75t15l2000 lf50w150r75t15l2500 lf50w150r75t15l3500 lf50w200r100t15l1000 lf50w200r100t15l2000 lf50w200r100t15l2500 lf50w200r100t15l3500 lf50w50r25t20l1000 lf50w50r25t20l2000 lf50w50r25t20l2500 lf50w50r25t20l3500 lf75w50r25t20l1000 lf75w50r25t20l2000 lf75w50r25t20l2500 lf75w50r25t20l3500 lf100w50r25t20l1000 lf100w50r25t20l2000 lf100w50r25t20l2500 lf100w50r25t20l3500 lf150w50r25t20l1000 lf150w50r25t20l2000 lf150w50r25t20l2500 lf150w50r25t20l3500 lf50w50r50t20l1000 lf50w50r50t20l2000 lf50w50r50t20l2500 lf50w50r50t20l3500 lf50w50r75t20l1000 lf50w50r75t20l2000 lf50w50r75t20l2500 lf50w50r75t20l3500 lf50w50r100t20l1000 lf50w50r100t20l2000 lf50w50r100t20l2500 lf50w50r100t20l3500 lf50w100r50t20l1000 lf50w100r50t20l2000 lf50w100r50t20l2500 lf50w100r50t20l3500 lf50w150r75t20l1000 lf50w150r75t20l2000 lf50w150r75t20l2500 lf50w150r75t20l3500 lf50w200r100t20l1000 lf50w200r100t20l2000 lf50w200r100t20l2500 lf50w200r100t20l3500

1.19 1.16 1.03 1.30 1.22 1.19 1.09 0.92 0.57 0.41 0.22 1.09 0.76 0.42 0.22 1.08 0.56 0.42 0.23 1.05 0.54 0.41 0.23 1.35 0.70 0.53 0.26 1.41 0.74 0.56 0.28 1.45 0.76 0.57 0.30 1.33 1.28 1.22 0.97 1.34 1.28 1.17 1.07 1.34 1.24 1.21 1.11

Y Y Y Y Y Y Y Y F F F Y F F F Y F F F Y F F F Y F F F Y F F F Y F F F Y Y Y F Y Y Y Y Y Y Y Y

F F F Y F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Y F F F Y F F F Y F F F

1.16 1.19 1.19 1.12 1.11 1.12 1.10 1.08 1.32 1.37 1.31 1.28 1.77 1.38 1.33 1.27 1.29 1.38 1.34 1.22 1.26 1.36 1.34 1.56 1.57 1.68 1.50 1.63 1.64 1.75 1.58 1.66 1.65 1.75 1.60 1.24 1.42 1.52 1.62 1.19 1.25 1.21 1.26 1.16 1.15 1.16 1.16

1.19 1.16 1.30 1.30 1.22 1.19 1.09 1.18 1.29 1.29 1.20 1.40 1.72 1.30 1.23 1.38 1.26 1.29 1.23 1.33 1.23 1.28 1.24 1.71 1.54 1.59 1.38 1.78 1.61 1.65 1.46 1.81 1.63 1.65 1.48 1.33 1.53 1.66 1.68 1.34 1.28 1.17 1.38 1.34 1.24 1.21 1.15

1.19 1.16 1.17 1.30 1.22 1.19 1.09 1.06 1.18 1.22 1.17 1.26 1.58 1.23 1.19 1.25 1.15 1.22 1.19 1.20 1.13 1.21 1.20 1.54 1.41 1.49 1.34 1.60 1.47 1.55 1.41 1.63 1.49 1.55 1.43 1.33 1.38 1.49 1.51 1.34 1.28 1.17 1.24 1.34 1.24 1.21 1.11

1.26 1.26 1.20 1.31 1.25 1.23 1.17 1.09 1.10 1.17 1.22 1.29 1.47 1.18 1.25 1.28 1.08 1.18 1.25 1.23 1.05 1.16 1.26 1.58 1.32 1.43 1.39 1.65 1.38 1.48 1.47 1.69 1.39 1.47 1.48 1.37 1.46 1.50 1.45 1.36 1.35 1.28 1.27 1.35 1.27 1.27 1.22

1.21 1.19 1.08 1.30 1.23 1.20 1.12 0.97 1.03 1.17 1.22 1.16 1.38 1.18 1.25 1.14 1.00 1.18 1.25 1.10 0.98 1.16 1.26 1.42 1.21 1.43 1.39 1.49 1.25 1.48 1.47 1.53 1.25 1.47 1.48 1.34 1.33 1.31 1.11 1.35 1.30 1.21 1.14 1.34 1.25 1.23 1.14

120 1.36 0.12 0.91

120 1.37 0.12 0.85 3.52 0.85 3.52

120 1.28 0.10 0.85 3.38 0.85 3.38

120 1.29 0.09 0.80 3.90 0.85 3.46

120 1.20 0.11 0.80 3.50 0.85 3.07

Number of data Mean (Pm) COV (Vp) Resistance factor (Фo) Reliability index (βo) Resistance factor (Ф1) Reliability index (β1)

0.85 3.49

*Y=Yielded failure; **F=Flexure failure.

is found to be ~94–110% for 200% increase in lf (from lf=50 mm). 4) In general, Pu increases with increase in w, however the rate of increase seems to be relatively higher for w≤100 mm, especially for l≥2000 mm. 5) A near linear increase in Pu with increase in thickness is observed, with the rate of increase of t for all height of columns but decreases with increase in l. 6) It can be observed that, although the decrease in Pu with l is quite expected (due to increase in global/member slenderness), the rate of decrease is relatively less sensitive (~74–77%) to variations in lf and r; however, it is quite sensitive (~8–77%) to variation in w. 7) All the compared design codes, method and their modified versions are reliable and applicable, with βo value of 3.28, 3.52, 3.90 for EN1993-1-4 [35], AS/NZS 4673 [36] and DSM [37,54]); and 3.38, 3.50

thickness and columns heights. Based on the parametric study, the following conclusions have been drawn: 1) Three failure or deformation mechanisms have been identified viz., 1) Yield to yield with local bucking (Mechanism ST1), 2) Yield to yield with flexural buckling (Mechanism ST2), and 3) Flexural to flexural buckling (Mechanism ST3). ST1, ST2 and ST3 mechanisms corresponds to λ < 0.32; 0.32≤λ ≤1.0; and λ > 1.0 respectively. Both ST1 and ST2 corresponds to yield initiated failure mechanism at Pu, whilst ST3, corresponds to flexure buckling initiated mechanism. 2) It is seen that semi-circular curve section (r/w=0.5) showed the highest load capacity, with Pu remaining almost flat for r/w≥1.0. 3) A near linear increase in Pu with lf is observed; the average increase 58

Thin-Walled Structures 113 (2017) 47–60

K. Sachidananda, K.D. Singh

(a)

(b)

1.5

EN 1993-1-4

2.0

Mechanism-ST2

1.0

1.5

Mechanism-ST3

Pu/PEN

Pu/Agfy

Mechanism-ST1

Euler

1.0

0.5 EN 1993-1-4

0.5

0.0

0.0 0

1

_

2

3

4

0

1

3

2

_

Ȝ

Fig. 14. Comparison of EN 1993-1-4 [35] and FE results for t=10–20 mm.

Fig. 15. Comparison of AS/NZS 4673 [36], modified AS/NZS 4673 (Huang and Young [9]) and FE results for t=10–20 mm.

(a) 1.5

Modified DSM (Huang and Young, 2014) DSM

1.2

(b)

2.0

0.9

Mechanism-ST2

Pu/PDSM

Pu/Py

Mechanism-ST1

Mechanism-ST3

0.6

1.0 Modified DSM (Huang and Young, 2014) DSM

0.3 0.0

0

1

2

3

0.0

4

0

1

2

3

c

c

Fig. 16. Comparison of DSM, modified DSM (Huang and Young [9]) and FE results for t=10–20 mm. [2] A. Mann, The structural use of stainless steel, Struct. Eng., 71(4), 60–68. [3] L. Gardner, The use of stainless steel in structures, Prog. Struct. Eng. Mater. 7 (2) (2005) 45–55. [4] M. Ashraf, L. Gardner, D.A. Nethercot, Compression strength of stainless steel crosssections, J. Constr. Steel Res. 62 (1–2) (2006) 105–115. [5] M. Theofanous, L. Gardner, Experimental and numerical studies of lean duplex stainless steel beams, J. Constr. Steel Res. 66 (2010) 816–825. [6] N. Saliba, L. Gardner, Experimental study of the shear response of lean duplex stainless steel plate girders, Eng. Struct. 46 (2013) 375–391.

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